Maybe for child support, but not for US or British nationality then or now. A recent US court refused to recognize the US citizenship without nationalization of a foreign-born child whose mother was an unmarried foreign national and whose US citizen father later acknowledged the child.
The 1948 BNA has similar rules for the children of unmarried or bigamously married foreign women who bear children in foreign countries to British citizens.
Paternity does not convey legal citizenship meaning nationality meaning subject to sovereignty.
See Miller v. Albright:
As I said, the fact that Barack Obama’s father was not a United States citizen is sufficient to make the case that he was not a natural-born citizen.
The rest is superfluous ,,, where he was born, what citizenship his father held, how old his mother was, or even whether or not his parents were married. There is ample documentation of legal paternity.